Core-valence effective potential

This form for the one-electron matrix elements was suggested through an expansion of of Eq. [14b], The third term of Eq. [29a] and the third term of [29c] represent the two-center part of a core-valence effective potential introduced to compensate for the repulsive nature of the neglected core orbitals in a valence orbital only treatment. 

The simplicity of the alkali diatomic molecules (equivalently the interactions of two alkali metal atoms) is illustrated by the work of Konowalow. and discussed in his contribution to this volume (33). It should be noted that because of the approximate separation of core and valence electron motion in these species, fully ab initio treatments are simpler (only valence electron excitations contribute significantly in a configuration interaction treatment) and replacement of the alkali ion cores by effective potentials becomes an accurate approximation. 

Valence orbital Xij is the lowest energy solution of equation 9.23 only if there are no core orbitals with the same angular momentum quantum number. Equation 9.23 can be solved using standard atomic HF codes. Once these solutions are known, it is possible to construct a valence-only HF-like equation that uses an effective potential to ensure that the valence orbital is the lowest energy solution. The equation is written... 

To date, the only applications of these methods to the solution/metal interface have been reported by Price and Halley, who presented a simplified treatment of the water/metal interface. Briefly, their model involves the calculation of the metal s valence electrons wave function, assuming that the water molecules electronic density and the metal core electrons are fixed. The calculation is based on a one-electron effective potential, which is determined from the electronic density in the metal and the atomic distribution of the liquid. After solving the Schrddinger equation for the wave function and the electronic density for one configuration of the liquid atoms, the force on each atom is ciculated and the new positions are determined using standard molecular dynamics techniques. For more details about the specific implementation of these general ideas, the reader is referred to the original article. ... 

For the very highest accuracy, the effect of at least core-valence correlation should be explored. This must be accompanied by some serious effort to extend the basis so that core-correlating functions are included. Using valence-optimized basis sets and including core correlation is not only a waste of computer time, but a potential source of problems, as it can substantially increase BSSE. This point is not well appreciated the prevailing view appears to be that no harm can come of correlating the core when the basis set is inadequate. This is not so. 

The generalization of the pseudopotential method to molecules was done by Boni-facic and Huzinaga[3] and by Goddard, Melius and Kahn[4] some ten years after Phillips and Kleinman s original proposal. In the molecular pseudopotential or Effective Core Potential (ECP) method all core-valence interactions are approximated with l dependent projection operators, and a totally symmetric screening type potential. The new operators, which are parametrized such that the ECP operator should reproduce atomic all electron results, are added to the Hamiltonian and the one electron ECP equations axe obtained variationally in the same way as the usual Hartree Fock equations. Since the total energy is calculated with respect to this approximative Hamiltonian the separability problem becomes obsolete. 

The one-center approximation allows for an extremely rapid evaluation of spin-orbit mean-field integrals if the atomic symmetry is fully exploited.64 Even more efficiency may be gained, if also the spin-independent core-valence interactions are replaced by atom-centered effective core potentials (ECPs). In this case, the inner shells do not even emerge in the molecular orbital optimization step, and the size of the atomic orbital basis set can be kept small. A prerequisite for the use of the all-electron atomic mean-field Hamiltonian in ECP calculations is to find a prescription for setting up a correspondence between the valence orbitals of the all-electron and ECP treatments.65-67... 

The pseudopotential method relies on the separation (in both energy and space) of electrons into core and valence electrons and implies that most physical and chemical properties of materials are determined by valence electrons in the interstitial region. One can therefore combine the full ionic potential with that of the core electrons to give an effective potential (called the pseudopotential), which acts on the valence electrons only. On top of this, one can also remove the rapid oscillations of the valence wavefunctions inside the core region such that the resulting wavefunction and potential are smooth. 

The atttractive prospect of treating only the valence electrons in ab initio calculations, by devising an effective potential formulation for the core electrons, is not new, and the pseudopotential idea in Hartree-Fock calculations has been extensively explored.127 Kahn and Goddard128-129 have, however, shown how a unique and... 

Other complications are associated with the partitioning of the core and valence space, which is a fundamental assumption of effective potential approximations. For instance, for the transition elements, in addition to the outermost s and d subshells, the next inner s and p subshells must also be included in the valence space in order to accurately compute certain properties (54). A related problem occurs in the alkali and alkaline earth elements, involving the outer s and next inner s and p subshells. In this case, however, the difficulties are related to core-valence correlation. Muller et al. (55) have developed semiempirical core polarization treatments for dealing with intershell correlation. Similar techniques have been used in pseudopotential calculations (56). These approaches assume that intershell correlation can be represented by a simple polarization of one shell (core) relative to the electrons in another (valence) and, therefore, the correlation energy adjustment will be... 

Replacement of the orthogonality relation of core and valence electrons by an effective potential applied to the valence shell... 

Conventional shape-consistent effective potentials (67-70), whether relativistic or not, are typically formulated as expansions of local potentials, U (r), multiplied by angular projection cperators. The expansions are tnmcated after the lowest angular function not contained in the core. The last (residual) term in the expansion typically represents little more than the simple ooulombic interaction between a valence electron and the core (electrons and corresponding fraction of the nuclear charge) and is predominantly attractive. The lower A terms, on the other hand., include strongly... 

In the effective potential approximation Mg is isoelectronic with Be. But, as can be seen in Figure 2, the Mg REP is composed of three terms (s, p and d) with the s and p both repulsive. As a result, even though the correlation correction is almost as large as in Be the multi-determinant correction resulting from Equation 3 is only a tenth as big (see Table II). The discrepancy between values from references 2) and ( ) is due to large statistical or extrapolation error. Note that unlike Be one cannot make comparisons with experimental results without first taking core-valence... 

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